Step 1 · Biostatistics & Epidemiology

Test Characteristics

Name: Test Characteristics
Creator: Veli Bakalov, MD

Sensitivity, specificity, and how prevalence bends predictive value

Biostatistics & Epidemiology·Diagnostic Testing·5 min read·8 high-yield points·Reviewed by Veli Bakalov, MD @HemeOncBuddy

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Sensitivity = TP/(TP + FN) and specificity = TN/(TN + FP); both are intrinsic to the test and do not change with prevalence.
SnNout: a negative result on a highly sensitive test rules disease out; SpPin: a positive result on a highly specific test rules disease in.
PPV = TP/(TP + FP) rises with prevalence; NPV = TN/(TN + FN) falls as prevalence rises.
In a rare disease even an excellent test has a low PPV: a 99%-sensitive, 99%-specific test at 0.1% prevalence gives a PPV of about 9%.
Lowering the cutoff raises sensitivity and NPV but lowers specificity and PPV; raising it does the opposite.
LR+ = sensitivity/(1 − specificity) and LR− = (1 − sensitivity)/specificity; posttest odds = pretest odds × LR; LR+ above 10 or LR− below 0.1 is strong.
ROC curves plot sensitivity against 1 − specificity; an AUC of 1.0 is perfect and 0.5 equals chance.
Screen with a sensitive test and confirm with a specific one; serial testing raises specificity, whereas parallel testing raises sensitivity.

The 2 × 2 table

Everything starts here. Build the table with the test result (positive or negative) as the rows and the true disease status (present or absent) as the columns. True positives (TP) and false positives (FP) sit in the top row, and false negatives (FN) and true negatives (TN) sit in the bottom row. Every test statistic comes from these four cells. Statistics read down the columns (sensitivity, specificity) describe the test itself. Statistics read across the rows (predictive values) describe what a result means for the patient.

Deriving test statistics

Result	Disease present	Disease absent	Row statistic
Test positive	TP (a)	FP (b)	PPV = TP/(TP + FP)
Test negative	FN (c)	TN (d)	NPV = TN/(TN + FN)
Column statistic	Sensitivity = TP/(TP + FN)	Specificity = TN/(TN + FP)	—

False-negative rate = 1 − sensitivity; false-positive rate = 1 − specificity; accuracy = (TP + TN)/total.

Sensitivity and specificity

Sensitivity. Sensitivity is the proportion of truly diseased people who test positive, TP/(TP + FN). If a test is positive in 90 of 100 diseased patients, its sensitivity is 90%, whatever the prevalence. A highly sensitive test has few false negatives, because the false-negative rate equals 1 − sensitivity. That makes it ideal for screening and for ruling disease out.

Specificity. Specificity is the proportion of truly healthy people who test negative, TN/(TN + FP). A highly specific test has few false positives, so it is ideal for confirming disease and ruling it in.

SnNout and SpPin. When a highly SeNsitive test is Negative, it rules OUT disease, because such a test almost never misses disease. This is why high-sensitivity tests are used first, when missing a dangerous diagnosis is unacceptable. When a highly SPecific test is Positive, it rules IN disease, because false positives are rare. Specific tests confirm the diagnosis after a sensitive screen is positive.

Sensitivity and specificity are intrinsic properties of the test at a given cutoff. They do not change with prevalence.

Predictive values and prevalence

Positive predictive value. PPV is the probability that a person with a positive test truly has the disease, TP/(TP + FP). It rises with higher prevalence and with higher specificity.

Negative predictive value. NPV is the probability that a person with a negative test is truly disease-free, TN/(TN + FN). It rises as prevalence falls, so in low-prevalence settings a negative test is very reassuring.

Prevalence moves PPV and NPV. As prevalence rises, PPV increases and NPV decreases, while sensitivity and specificity stay the same. In a rare disease, even an excellent test yields a low PPV because false positives from the large healthy group swamp the few true positives.

Worked example. Take a test that is 99% sensitive and 99% specific, used in 100,000 people with a prevalence of 0.1%. Of the 100 diseased people, 99 test positive. Of the 99,900 healthy people, 1% (999) also test positive. Only 99 of the 1,098 positives are real.

≈9%PPV of a 99%-sensitive, 99%-specific test at 0.1% prevalence

>99.99%NPV in the same population

This is why a positive screen in a low-prevalence population is confirmed with a more specific test before anyone is labeled as diseased.

Likelihood ratios and posttest probability

Definitions. Likelihood ratios combine sensitivity and specificity into one number and, like them, do not depend on prevalence.

LR+ = sensitivity / (1 − specificity): how much a positive result raises the odds of disease.
LR− = (1 − sensitivity) / specificity: how much a negative result lowers the odds of disease.
An LR+ above 10 or an LR− below 0.1 changes disease probability strongly. An LR of 1 means the test is useless.

Pretest probabilityEssentially the prevalence in the patient's population, adjusted for the clinical picture
Convert to pretest oddsOdds = p / (1 − p)
Multiply by the LRPosttest odds = pretest odds × LR
Convert backPosttest probability = odds / (1 + odds)

Checking the example. The 99%/99% test has an LR+ of 99. A pretest probability of 0.1% (odds ≈ 0.001) becomes posttest odds of about 0.099, a probability of about 9%. That matches the PPV from the 2 × 2 table. The Fagan nomogram performs the same conversion graphically.

Cutoffs, ROC curves and screening

Moving the cutoff. Lowering the diagnostic threshold labels more people positive. For example, a lower glucose cutoff for diabetes raises sensitivity and NPV but lowers specificity and PPV, because it creates more false positives. Raising the threshold does the opposite. Screening tests use a low threshold to maximize sensitivity, and confirmatory tests use a high threshold for specificity.

What moves each statistic

Change	Sensitivity	Specificity	PPV	NPV
Lower the cutoff (more positives)	↑	↓	↓	↑
Raise the cutoff (fewer positives)	↓	↑	↑	↓
Higher prevalence	No change	No change	↑	↓
Lower prevalence	No change	No change	↓	↑

ROC curve. A receiver operating characteristic curve plots the true-positive rate (sensitivity) against the false-positive rate (1 − specificity) across all possible cutoffs. The area under the curve (AUC) measures overall discrimination. An AUC of 1.0 is perfect, and 0.5 lies on the diagonal and is no better than a coin flip. A test with an AUC of 0.92 discriminates better than one with 0.68. The point nearest the upper-left corner offers the best balance of sensitivity and specificity.

Screening principles (Wilson-Jungner). Screening makes sense for a disease that is common and serious, has a detectable asymptomatic phase and has effective early treatment. The test should be cheap, safe, sensitive and acceptable. Screen with a sensitive test, then confirm with a specific one. Testing in series (both tests must be positive) raises specificity and PPV. Testing in parallel (either test positive counts) raises sensitivity and NPV.